Topological Two-Dimensional Floquet Lattice on a Single Superconducting Qubit

PHYSICAL REVIEW LETTERS 126, 163602 (2021)

Daniel Malz 1,2,* and Adam Smith3,4,5,* 1Max Planck Institute for Quantum Optics, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany 2Munich Center for Quantum Science and Technology, Schellingstraße 4, D-80799 München, Germany 3Department of Physics, TFK, Technische Universität München, James-Franck-Straße 1, D-85748 Garching, Germany 4School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom 5Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, United Kingdom (Received 11 December 2020; revised 1 March 2021; accepted 22 March 2021; published 23 April 2021)

Current noisy intermediate-scale quantum (NISQ) devices constitute powerful platforms for analog quantum simulation. The exquisite level of control offered by state-of-the-art quantum computers make them especially promising to implement time-dependent Hamiltonians. We implement quasiperiodic driving of a single qubit in the IBM Quantum Experience and thus experimentally realize a temporal version of the half-Bernevig-Hughes-Zhang Chern insulator. Using simple error mitigation, we achieve consistently high fidelities of around 97%. From our data we can infer the presence of a topological transition, thus realizing an earlier proposal of topological frequency conversion by Martin, Refael, and Halperin. Motivated by these results, we theoretically study the many-qubit case, and show that one can implement a wide class of Floquet Hamiltonians, or time-dependent Hamiltonians in general. Our study highlights promises and limitations when studying many-body systems through multifrequency driving of quantum computers.

DOI: 10.1103/PhysRevLett.126.163602

Introduction.—Noisy intermediate-scale quantum simulation [9,10], which potentially incurs significantly less (NISQ) computers may not yet offer fully fault-tolerant overhead. This perspective has been explored in a series of quantum computing facilities, but they nevertheless con- theoretical and experimental works [11–15]. If the intrinsic stitute a versatile experimental platform with the potential many-body nature of quantum computers is combined with for fundamental research, small-scale computation or the capacity to apply essentially arbitrary drives, they may quantum simulation [1]. The typical model of a quantum serve also as powerful analog quantum simulators for very computer is that of a quantum circuit, which is a sequence large classes of time-dependent Hamiltonians. of gates applied to the qubits [2]. In principle, the time The evolution under time-dependent Hamiltonians is evolution of any many-body quantum systems can be incredibly rich and exhibits many novel phenomena, even simulated by applying a Trotterization, which turns con- at the level of individual qubits. A particular example is the tinuous time evolution into a discrete local quantum circuit temporal topological transition that occurs in the presence [3]. This results in a digital quantum simulation, which has of quasiperiodic driving, theoretically predicted by Martin, been benchmarked for a range of different models on Refael, and Halperin in 2017 [16]. Using a Floquet treat- existing quantum computers [4–7]. ment of the driven qubit, the dynamics is related to the In superconducting circuits, the currently leading tech- properties of a two-dimensional lattice model, the half- nology, quantum circuits are constructed from a set of Bernevig-Hughes-Zhang (BHZ) Chern insulator [17].Asa available gates, which correspond to a set of carefully function of time, the driven qubit explores the whole calibrated microwave pulses applied to its input ports [8]. Brillouin zone, which causes the work done by the two The abstraction into quantum circuits hides the complexity drives to be quantized and proportional to the integer Chern of the underlying many-body system, whose continuous number, which is determined by the parameters in the evolution offers exciting directions in analog quantum drives. Recently, temporal topology has been classified, analogously to the classification of topological insulators [18], and extension to larger systems, such as spin- Published by the American Physical Society under the terms of resonator systems [19], or two-qubit systems with inter- the Creative Commons Attribution 4.0 International license. actions [20] have been proposed. While quasiperiodic Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, driving typically maps to systems without boundary, one and DOI. Open access publication funded by the Max Planck can, in principle, also introduce boundaries through quan- Society. tum feedback [21].

0031-9007=21=126(16)=163602(7) 163602-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 126, 163602 (2021)

In this work we experimentally demonstrate this tem- rotating with respect to the time-dependent Hamiltonian ω 2 − σ poral topological behavior of a single qubit on an existing HzðtÞ¼ð 0= hzðtÞÞ z, we rewrite Eq. (1) as [32] quantum device, using continuous driving, implemented ⃗ σ⃗ with the fine-grained access offered by QISKIT pulse [22]. HspinðtÞ¼hðtÞ · : ð4Þ Choosing a specific driving with two incommensurate frequencies, we observe a topological transition in the Note the subtlety in the chosen rotating frame. Instead of temporal dynamics of the qubit, finding good agreement changing the qubit frequency in the lab frame, we repar- with simulations. Despite achieving high fidelities of ametrize time by passing into a rotating frame with respect around 97% after error mitigation, the Chern number to HzðtÞ. This constitutes a continuous version of virtual Z inferred from the measured frequency conversion shows gates [35]. much larger errors. We develop a simple noise model that Topological frequency conversion.—Given Eq. (4),we explains and reproduces this effect. can now realize the proposal by Martin et al. [16]. We apply Motivated by this experiment, we theoretically derive the the quasiperiodic time-dependent magnetic field class of Hamiltonians that can readily be implemented on η ω ϕ σ ω ϕ σ state-of-the-art quantum computers. As one concrete exam- HðtÞ¼ fsinð 1t þ 1Þ x þ sinð 2t þ 2Þ y ple, this offers an exciting perspective to study strongly þ½M − cosðω1t þ ϕ1Þ − cosðω2t þ ϕ2Þσzg; ð5Þ interacting Floquet systems [23–27] with an exquisite level of control. Site-selective control as well as high-fidelity ω ω where the ratio 2= 1 shouldpffiffiffi be irrational. In the follow- single-site readout confers quantum computers certain ing, we set ω2=ω1 ¼ð1 þ 5Þ=2. We consider the model advantage over other quantum simulators based on light in the strong drive limit, i.e., η ≫ ω1; ω2. In this limit, a [28] or ultracold atoms [29], making them ideally suited for Floquet ansatz reveals a direct connection to the two- the analog quantum simulation of generic many-body time- dimensional half-BHZ Chern insulator [17] with a constant dependent Hamiltonians. electric field applied [16,32]. In the strong-drive limit, the — Theoretical description. In quantum computers based electric field is weak and leads to a slow adiabatic on superconducting transmonlike qubits [30,31], the evolution of the initial wavepacket through the Brillouin Hamiltonian describing a single qubit can be cast in the zone. During this evolution, which explores the whole form of a Duffing oscillator with driving Brillouin zone, the system effectively measures the Chern number, which results in a topologically quantized energy ω † † † † HðtÞ¼ 0a a þ Ua a aa þða þ a ÞDðtÞ: ð1Þ pumping rate from one drive to the other [16] ω The qubit frequency is denoted 0 and U quantifies the ðW1 − W2Þ anharmonicity that separates the lowest two levels that π ¼ C; ð6Þ ω1ω2T define the qubit from the higher levels of the superconducting circuit. The ideal drive signal is parametrized where Wi is the work done by the ith drive, defined below as [22] [Eq. (8)]. As a function of M, the system undergoes a topological transition in which the Chern number changes. Ω max ω Δ DðtÞ¼ Re½eið 0þ cÞtdðtÞ; ð2Þ To determine the pumping rate experimentally, we 2 measure the work done by each of the drives. If we first split the Hamiltonian into the two contributions from each where Ω denotes the maximum Rabi frequency attain- max drive, able in the system, and ω0 þ Δc is the carrier frequency. In the following, we choose the detuning Δc ¼ 0. HðtÞ¼h1ðtÞþh2ðtÞþηMσ ; ð7Þ Anticipating our later interpretation as a spin-1=2 particle z in a time-dependent field, we parameterize the dimension- then the work done by each drive over a period T is given less drive shape d t ∈ C; d t ≤ 1 in terms of the ð Þ j ð Þj by dimensionless magnetic field h˜ ðtÞ¼h˜ ðtÞþih˜ ðtÞ þ x y Z Z T dh ðtÞ t Ψ i Ψ ≡ ˜ ϕ ϕ −2Ω ˜ 0 0 WiðTÞ¼ dth ðtÞj j ðtÞi; ð8Þ dðtÞ hþðtÞ exp i ðtÞ; ðtÞ¼ max hzðt Þdt : 0 dt 0 ð3Þ where jΨðtÞi is the state of the qubit at time t evolving under the time-dependent Schrödinger equation To treat the system as a qubit, the maximum drive i∂tjΨðtÞi ¼ HðtÞjΨðtÞi, where jΨðt ¼ 0Þi is chosen to be strength needs to be much weaker than the anharmonicity an instantaneous eigenstate of Hðt ¼ 0Þ. Ω ≪ max U. Assuming this is fulfilled, applying a Experiment.—Our experimental protocol consists of four rotating-wave approximation and passing into a frame main pulse sequences. First, we initialize the qubit in the

163602-2 PHYSICAL REVIEW LETTERS 126, 163602 (2021) instantaneous eigenstate of Hamiltonian Eq. (5) at t ¼ 0. This is achieved using an IBM calibrated pulse sequence to perform a general qubit rotation. Second, the main pulse sequence implements the time-dependent Hamiltonian (5), with the drive parametrized as in Eq. (2). We perform experiments with different drive lengths of up to 20 μsto obtain 800 data points. Third, we apply an IBM calibrated pulse to change the basis. For each drive length, we rotate into the X, Y, and Z bases in order to perform full state tomography. Finally, we perform single-shot projective measurements of the qubit using an IBM pre-calibrated readout pulse sequence. We average over 8192 shots for each observable corresponding to a statistical error of approximately 1%. pffiffiffi We fix ω2 ¼ φω1, where φ ¼ð1 þ 5Þ=2 is the golden ratio. Ideally, we would be like to set the ratio ω1=η as small as possible, to get as close as possible to adiabatic evolution. However, due to the finite coherence time τ ≳ 100 μs of the IBM Q device, we must choose −1 FIG. 1. Tomography data for M ¼ 1, ω1 ¼ 0.125. The top three ω1 ≪ τ. In terms of the maximum Rabi frequency Ω η 0 9Ω panels compare experimentally measured Pauli expectation max, we choose ¼ . max, in order to avoid driving values against exact numerical simulation. We also show the transitions to higher excited states. Using numerical sim- purity of the measured state, which we fit by the function ulations we found the best compromise was to choose a 1=2 þ ae−t=λ, where a ¼ 0.387 and λ ≈ 109 μs is compatible total simulation time of 20 μs and set ω1 ¼ 0.125η, which with the device coherence time as measured by IBM. Note that −1 corresponds to ω1 ≈ 240 ns. In order to improve the a<0.5 as the measurement sequence take a finite time, such that fidelity of our results we start the drive with a linear ramp the qubit has lost purity even when the simulation time is zero. F ψ ϕ 2 of the drive frequencies ω1, ω2 over a period of 444 ns. This The bottom panel shows the fidelity ¼jh j ij between the reduces transient effects and reduces high-frequency Rabi measured state projected onto the Bloch sphere, jψi, and the ϕ −t=ξ oscillations seen in the simulation results. The single-qubit numerically simulated state, j i. We fit the fidelity with ae , where a ≈ 0.99 and ξ ≈ 2.71 ms, which verifies the effectiveness IBM device we used, codenamed armonk, had qubit of error mitigation by projecting onto the Bloch sphere. frequency ω0 ≈ 4.97 GHz and maximum Rabi frequency Ω ≈ 36 9 of max . MHz. With the above experimental protocol we measure the The experimental tomography data is shown in Fig. 1 for observables in the frame rotating with the qubit frequency. M ¼ 1, which closely matches the exact numerical simu- Since the Hamiltonian Eq. (4) is only realized in a given lation. From this data we can also compute the purity of the time-dependent reference frame, we must additionally measured state Trðρ2Þ, also shown in Fig. 1. The measured perform a virtual-Z rotation, which we achieve by post- purity is less than one due to decoherence over the course of processing the data to apply the rotation the experiment along with the significant ∼3% measurement error. Fitting the purity with an exponential decay, we σ ϕ σ ϕ σ extract a decoherence time of λ ≈ 109 μs (at M ¼ 1), which h xirotating ¼ cos ðtÞh xiþsin ðtÞh yi; is consistent with the T1;T2 ≳ 100 μs decoherence times σ − ϕ σ ϕ σ h yirotating ¼ sin ðtÞh xiþcos ðtÞh yi; ð9Þ measured by IBM. Results.—To obtain the work done by each drive [Eq. (8)], ⃗ where ϕðtÞ is given in Eq. (3). When post-processing the we first compute hdhiðtÞ=dti¼dhiðtÞ=dt · hσ⃗i from the data, we additionally mitigate some of the errors due to data. We then perform the integration in Eq. (8) numerically. decoherence and measurement error by projecting the Figure 2 shows the experimental results for the work done, measured qubit density matrix onto the Bloch sphere. for the case of M ¼ 1. This corresponds to a phase of the We find that this method of error mitigation improves BHZ model with Chern number C ¼ −1, resulting in an quantitative agreement with numerical simulations in all average linear decrease (increase) of W1 (W2). The exper- cases we considered. This effectively removes the effects of imental results are in good quantitative agreement with the dominant depolarizing channel [36]. With this error numerical simulations. Furthermore, by fitting a linear curve mitigation strategy, the average fidelity across all experi- to this data using least-squares regression, the slope is in ments is 0.971 (excluding M ¼ 1.7, 2.3 due to strong close agreement to that predicted by Eq. (6). diabatic effects that arise due to the proximity to the gap Figure 3 shows the extracted Chern number for a range closing at M ¼ 2 [32]). of values of M, compared against numerical simulations of

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FIG. 2. Work done by the two incommensurate drives, calcu- FIG. 3. Experimentally extracted Chern number as a function of lated using Eq. (8). The experimental data for M ¼ 1, ω1 ¼ 0.125 M. The Chern number is extracted from the linear least-squares measured at 800 points is compared with a numerical simulation regression fit to the work done by each drive, and using Eq. (6). of the same setup. The experimental data is fitted with a line The error bars correspond to the 95% confidence intervals for the (colored dashed lines) using least-squares regression, for which fit. The experimental data are compared against the numerical the 95% confidence interval for the slope is shown as the colored simulation of the same setup (blue dots) and with the exact value region. The expected slope is shown as black dashed lines. of the Chern number for the corresponding phase of the BHZ model (dashed black curve). The shaded blue region corresponds to a simple heuristic error model based on the average fidelity the same setup as well as the ideal result in the strong-drive 0.971 of the error mitigated experimental data [32]. This error limit. We find reasonable quantitative agreement with the model confirms that the comparatively large deviations of the simulations. Furthermore, the transition between different measured Chern number that we observe are produced already ≈3% phases with different Chern number is clearly visible in the from low loss in fidelity ( ). Insets show the distribution of 1 experimental data. Beyond the extracted value of the Chern experimental data on the Bloch sphere under dynamics for M ¼ and M ¼ 3, illustrating qualitative differences of the time number, the qualitative difference between the phases with evolution in each of the two phases jMj < 2 and jMj > 2. jMj < 2 and jMj > 2 is clearly seen by considering the portion of the Bloch sphere covered under time evolution, shown inset Fig. 3.ForjMj < 2 we observe that the state of between two qubits can be engineered in many ways, as the qubit explores the full Bloch sphere under the dynamics one can choose between direct capacitive or inductive of Eq. (5) reflecting that the surface traced by the time coupling, or indirect coupling via a resonator [37–41]. dependent state has nontrivial winding around the origin Indeed, alternative approaches have been taken by IBM and hence non-zero Chern number. When jMj > 2 the state (see, e.g., Ref. [42]) and Google [43], with each imple- of the qubit is instead restricted to either the north or south mentation having their own (dis-)advantages. hemisphere of the Bloch sphere and does not wind around We consider two cases in particular. The first is when the origin and the Chern number is zero. both qubit frequencies and their coupling is fixed, which While overall the agreement is good, the error in the is relevant for the devices developed by IBM. In this measured Chern number exceeds the error predicted by the case, the coupling has to be engineered with time- fidelity considerably. We find that this can be explained by dependent driving of the qubits in order to implement a simple error model, in which we take the state predicted in resonant processes to second order in the Hamiltonian the ideal scenario and randomly perturb its direction on the [44–46]. This technique produces tuneable XZ; YZ; ZX, Bloch sphere to reproduce the measured distribution of and ZY interactions, which are used to engineer CR fidelities with average 0.971 [32]. Note that the perturba- gates [45,47]. Implementing a similar single-qubit driv- tions at data points corresponding to different times are ing as before, and passing into a time-dependent refer- independent, as they correspond to different experimental ence frame, we derive an effective Hamiltonian of the runs. This perturbation translates into an error in the form [32] measured Chern number, whose standard deviation we X plot as a shaded region in Fig. 3. This shows that a error rate ð2Þ σðiÞσðjÞ Hint ¼ gijðtÞ z þ þ H:c: ð10Þ consistent with experimental data explains the large errors hiji in the extracted Chern number. Extension to qubit arrays.—In view of the vast research On bipartite lattices, a (virtual) rotation of every second effort in Floquet matter, the question is pertinent whether spin maps this to either an Ising interaction or (in general the presented scheme generalizes to several qubits. To anisotropic) XY interactions, making this technique very answer this question we must be aware that the coupling versatile. A drawback is that the Hamiltonian (10) is

163602-4 PHYSICAL REVIEW LETTERS 126, 163602 (2021) obtained to second order in the original Hamiltonian and of the Chern number in the same model using a nitrogen by neglecting quickly rotating terms. It is thus an vacancy centre. They determined the Chern number by approximation and care has to be taken that all the steps measuring the Berry curvature directly, whereas here we in its derivation are valid. Nevertheless, these conditions extract the Chern number from the topological pumping as can usually be fulfilled through careful choice of the originally proposed. As a result, they did not observe the driving parameters. It may further be possible to actively substantial errors in the extracted Chern number that we counteract unwanted effects from this approximation, found. Nevertheless, as we have demonstrated, the tempo- analogous to the DRAG scheme used to improve the ral topological nature of the qubit time evolution can clearly fidelity of digital quantum gates [8]. be extracted despite experimental shortcomings. With the If tunable interactions are available and the qubits can current progress in superconducting-qubit technology, we be brought into resonance [43], the driving scheme is expect that our understanding and the fidelity of available simplified and one readily arrives at the general spin quantum computers to increase rapidly, allowing for more Hamiltonian complex experiments, in particular with more qubits. X X Going beyond (quasi)periodic driving, the capability ⃗ðiÞ σ⃗ðiÞ σðiÞσðjÞ σðiÞσðjÞ HarrayðtÞ¼ h ðtÞ · þ gijðtÞð x x þ y y Þ to engineer time-dependent many-body Hamiltonians i hiji offers many exciting perspectives to investigate nonequilibrium physics. For example, ramping through a ð11Þ quantum phase transition might allow one to study with a magnetic field along z that takes the form Kibble-Zurek scaling or in general the dynamics of phase transitions such as the superfluid to Mott insulator ðiÞ transition [54]. Slow variation could also be used to h ðtÞ¼h ðtÞþω ðtÞ − ω0; ð12Þ z z i explore adiabatic algorithms and departures from them. Many-body non-equilibrium physics is notoriously dif- where ω are the qubit frequencies and ω0 is some i ficult to study with classical computers and this is arbitrarily chosen reference frequency. We note that tunable therefore a prime area of applicability for quantum qubit frequencies feature in many implementations simulators. Noisy intermediate-scale quantum computers [48–51], but not in all [52]. In the latter case—of fixed qubit frequencies but tunable interactions—one option offer a versatile combination of single-site control and might be to drive each qubit with a far off-resonant drive readout, and large enough system sizes, and thus promise to induce an drive-strength-dependent ac Stark shift. to support and complement analytical and computational When taking into account the second excited state of each approaches to understand many-body nonequilibrium qubit, the qubit array Hamiltonian (11) maps to a Bose- physics. As we have demonstrated with the single-qubit Hubbard model with time-dependent hopping and freely experiment, this is a realistic outlook. tuneable site-dependent drive and disorder. We note that the All of the data presented in this Letter, along with the interpretation as a (time-independent) Bose-Hubbard model python code used to generate these data and the figures in has enabled the experimental measurement of microscopic this Letter, can be found in a public repository [55]. features of the many-body localized phase [13].Movingto D. M. acknowledges funding from ERC Advanced Grant periodically varying hoppings could allow one to study QUENOCOBA under the EU Horizon 2020 program many-body Floquet models and implement quench and ramp (Grant Agreement No. 742102). A. S. was supported by experiments from carefully prepared initial states. the European Research Council (ERC) under the European — Discussion. While our experiment showcases some of Union’s Horizon 2020 research and innovation program the promises of analog quantum simulation with time- (grant agreement No. 771537), and in the latter stages of dependent Hamiltonians, it also highlights some this work by a Research Fellowship from the Royal of the difficulties that need to be overcome on the way. Commission for the Exhibition of 1851. The views Concretely, the experimental results shown in Fig. 3 would expressed are those of the authors and do not reflect the improve with greater coherence time of the qubit, as this official policy or position of IBM or the IBM Quantum ω would allow us to reduce the modulation frequencies 1 Experience team. and ω2 in the Hamiltonian, which in turn would improve the strong-driving and adiabatic-modulation approxima- tions. This is particularly important close to the topological transition at M 2 where the gap closes and adiabatic * ¼ These authors contributed equally to this work. evolution thus requires increasingly long timescales. [1] J. Preskill, Quantum computing in the NISQ era and In this specific experiment, we also encountered the beyond, Quantum 2, 79 (2018). problem that the measured experimental signature is [2] M. A. Nielsen and I. L. Chuang, Quantum Computation sensitive to even small amounts of noise. Recently, and Quantum Information (Cambridge University Press, Boyers et al. [53] demonstrated experimental measurement Cambridge, 2010).

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